Numerical study of specimen with steel inclusion: Influence of interfacial transition zone

ScienceDirect Available online at www.sciencedirect.com Procedia Structural Integrity 42 (2022) 1000–1007 2452-3216 © 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the 23 European Conference on Fracture – ECF23 10.1016/j.prostr.2022.12.126 10.1016/j.prostr.2022.12.126 2452-3216 © 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the 23 European Conference on Fracture – ECF23 Available online at www.sciencedirect.com ^ĐŝĞŶĐĞŝƌĞĐƚ Structural Integrity Procedia 00 (2019) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 23 European Conference on Fracture - ECF23 Numerical study of specimen with steel inclusion: Influence of interfacial transition zone Michal Vyhlídala,*, Jan Klusákb aBrno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic bCzech Academy of Sciences, Institute of Physics of Materials, Žižkova 22, 616 00 Brno, Czech Republic Abstract In this paper, the influence of the interfacial transition zone (ITZ) between steel inclusion and fine-grained cement-based composite on fracture behaviour is investigated. Specimens of the nominal dimensions 40 × 40 × 160 mm with the steel inclusion of the prismatic shape with nominal dimensions 8 × 8 × 40 mm were provided with an initial central edge notch with a depth 12 mm. The aim of this paper is to analyse the behaviour of such specimen by means of numerical modelling by finite element method (Ansys software). A simplified 2D model (plane strain) based on the fracture test configuration was created. The crack propagation assessment was based on the criterion of the average value of tangential stress 𝜎𝜎θθ calculated over certain distance d in dependence on the polar coordinate θ. It is assumed that the crack is initiated when the average stress 𝜎𝜎θθ(𝜃𝜃0) reaches its critical value 𝜎𝜎θθ,c(𝜃𝜃0), which depends on the fracture toughness 𝐾𝐾Ic of the material and on the distance d. From the detailed numerical analysis of the described fracture test, we concluded that the crack propagation path depends strongly on the fracture properties of ITZ. The central inclusion works as an obstacle to crack propagation only in the initial stage of failure. The mechanical fracture parameters of the ITZ strongly influence the overall fracture behavior of test specimens. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 Keywords: : Steel inclusion, fracture test, numerical simulation, crack propagation path, the criterion of average tangential stress. * Corresponding author. E-mail address: [email protected] Available online at www.sciencedirect.com ^ĐŝĞŶĐĞŝƌĞĐƚ Structural Integrity Procedia 00 (2019) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 23 European Conference on Fracture - ECF23 Numerical study of specimen with steel inclusion: Influence of interfacial transition zone Michal Vyhlídala,*, Jan Klusákb aBrno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic bCzech Academy of Sciences, Institute of Physics of Materials, Žižkova 22, 616 00 Brno, Czech Republic Abstract In this paper, the influence of the interfacial transition zone (ITZ) between steel inclusion and fine-grained cement-based composite on fracture behaviour is investigated. Specimens of the nominal dimensions 40 × 40 × 160 mm with the steel inclusion of the prismatic shape with nominal dimensions 8 × 8 × 40 mm were provided with an initial central edge notch with a depth 12 mm. The aim of this paper is to analyse the behaviour of such specimen by means of numerical modelling by finite element method (Ansys software). A simplified 2D model (plane strain) based on the fracture test configuration was created. The crack propagation assessment was based on the criterion of the average value of tangential stress 𝜎𝜎θθ calculated over certain distance d in dependence on the polar coordinate θ. It is assumed that the crack is initiated when the average stress 𝜎𝜎θθ(𝜃𝜃0) reaches its critical value 𝜎𝜎θθ,c(𝜃𝜃0), which depends on the fracture toughness 𝐾𝐾Ic of the material and on the distance d. From the detailed numerical analysis of the described fracture test, we concluded that the crack propagation path depends strongly on the fracture properties of ITZ. The central inclusion works as an obstacle to crack propagation only in the initial stage of failure. The mechanical fracture parameters of the ITZ strongly influence the overall fracture behavior of test specimens. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 Keywords: : Steel inclusion, fracture test, numerical simulation, crack propagation path, the criterion of average tangential stress. * Corresponding author. E-mail address: [email protected] Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 1001 Available online at www.sciencedirect.com ^ĐŝĞŶĐĞŝƌĞĐƚ Structural Integrity Procedia 00 (2019) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 23 European Conference on Fracture - ECF23 Numerical study of specimen with steel inclusion: Influence of interfacial transition zone Michal Vyhlídala,*, Jan Klusákb aBrno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic bCzech Academy of Sciences, Institute of Physics of Materials, Žižkova 22, 616 00 Brno, Czech Republic Abstract In this paper, the influence of the interfacial transition zone (ITZ) between steel inclusion and fine-grained cement-based composite on fracture behaviour is investigated. Specimens of the nominal dimensions 40 × 40 × 160 mm with the steel inclusion of the prismatic shape with nominal dimensions 8 × 8 × 40 mm were provided with an initial central edge notch with a depth 12 mm. The aim of this paper is to analyse the behaviour of such specimen by means of numerical modelling by finite element method (Ansys software). A simplified 2D model (plane strain) based on the fracture test configuration was created. The crack propagation assessment was based on the criterion of the average value of tangential stress 𝜎𝜎θθ calculated over certain distance d in dependence on the polar coordinate θ. It is assumed that the crack is initiated when the average stress 𝜎𝜎θθ(𝜃𝜃0) reaches its critical value 𝜎𝜎θθ,c(𝜃𝜃0), which depends on the fracture toughness 𝐾𝐾Ic of the material and on the distance d. From the detailed numerical analysis of the described fracture test, we concluded that the crack propagation path depends strongly on the fracture properties of ITZ. The central inclusion works as an obstacle to crack propagation only in the initial stage of failure. The mechanical fracture parameters of the ITZ strongly influence the overall fracture behavior of test specimens. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 Keywords: : Steel inclusion, fracture test, numerical simulation, crack propagation path, the criterion of average tangential stress. * Corresponding author. E-mail address: [email protected] Available online at www.sciencedirect.com ^ĐŝĞŶĐĞŝƌĞĐƚ Structural Integrity Procedia 00 (2019) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 23 European Conference on Fracture - ECF23 Numerical study of specimen with steel inclusion: Influence of interfacial transition zone Michal Vyhlídala,*, Jan Klusákb aBrno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic bCzech Academy of Sciences, Institute of Physics of Materials, Žižkova 22, 616 00 Brno, Czech Republic Abstract In this paper, the influence of the interfacial transition zone (ITZ) between steel inclusion and fine-grained cement-based composite on fracture behaviour is investigated. Specimens of the nominal dimensions 40 × 40 × 160 mm with the steel inclusion of the prismatic shape with nominal dimensions 8 × 8 × 40 mm were provided with an initial central edge notch with a depth 12 mm. The aim of this paper is to analyse the behaviour of such specimen by means of numerical modelling by finite element method (Ansys software). A simplified 2D model (plane strain) based on the fracture test configuration was created. The crack propagation assessment was based on the criterion of the average value of tangential stress 𝜎𝜎θθ calculated over certain distance d in dependence on the polar coordinate θ. It is assumed that the crack is initiated when the average stress 𝜎𝜎θθ(𝜃𝜃0) reaches its critical value 𝜎𝜎θθ,c(𝜃𝜃0), which depends on the fracture toughness 𝐾𝐾Ic of the material and on the distance d. From the detailed numerical analysis of the described fracture test, we concluded that the crack propagation path depends strongly on the fracture properties of ITZ. The central inclusion works as an obstacle to crack propagation only in the initial stage of failure. The mechanical fracture parameters of the ITZ strongly influence the overall fracture behavior of test specimens. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 Keywords: : Steel inclusion, fracture test, numerical simulation, crack propagation path, the criterion of average tangential stress. * Corresponding author. E-mail address: [email protected] 2 Author name / Structural Integrity Procedia 00 (2019) 000–000 1. Introduction Concrete is a composite, which is formed, in a fresh state, from coarse aggregate, fine aggregate, binder (mostly Portland cement) and water, eventually admixtures. Due to the hydration process concrete obtain its valuable properties – high strength (mainly compressive), durability, fire resistance etc. During the hydration and hardening, the shrinkage and other processes occur and lead to the formation of micro and macro-defects (cracks, pores, etc.). Nevertheless, concrete structures are designed using procedures mentioned in standards – e. g. EN 1992-1-1 (2011) or recommendations which do not consider the existence of defects, such as cracks, pores, inclusions, transition zones, etc. However, these discontinuities can act as potential weak elements, which influences the fracture behavior of concrete structures. In the more complex design of these structures (e. g. complex geometry), it is advisable to apply the principles of fracture mechanics. In this paper, material discontinuity is formed by special steel inclusion placed in the middle of the test specimen. Such a specimen made of fine-grained cement-based composite was tested in three-point bending configuration. The aim of this paper is to identify the influence of the interface between steel inclusion and matrix which is formed by the interfacial transition zone (ITZ). In the paper, results of numerical assessment of crack propagation stages are presented. Nomenclature d averaging distance E Young’s modulus EITZ Young’s modulus of the interfacial transition zone Fappl applied loading force Fcrit critical value the applied force 𝐹𝐹𝑖𝑖𝑖𝑖(𝑛𝑛,𝜃𝜃) shape functions GMTS generalized maximum tangential stress H1 generalized stress intensity factor ITZ interfacial transition zone KI stress intensity factor in loading mode I KIc fracture toughness (under pure mode I) KIc, ITZ fracture toughness (under pure mode I) of the interfacial transition zone KIc, MTX fracture toughness (under pure mode I) of the matrix MTX matrix – fine-grained cement-based composite p1 stress singularity exponent 𝑟𝑟,𝜃𝜃 polar coordinates ν Poisson’s ratio 𝜎𝜎 ij stress tensor component 𝜎𝜎𝜃𝜃𝜃𝜃(𝑟𝑟,𝜃𝜃) tangential stress 𝜎𝜎𝜃𝜃𝜃𝜃(𝜃𝜃) average tangential stress 𝜎𝜎𝜃𝜃𝜃𝜃,c critical value of the average tangential stress 2. Theoretical background 2.1. The Interfacial Transition Zone The existence of the ITZ between aggregate and cement paste was first described in the 1950s by Farran (1956). The properties of the ITZ and its impact on the behavior of cement-based composites have been studied from many points of view since that. The ITZ is a region of about 50 µm in size, and its microstructure is formed mainly by ettringite needles and portlandite plates (Diamond and Huang, 1998). The significant feature of the ITZ is mainly 1002 Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 Author name / Structural Integrity Procedia 00 (2019) 000–000 3 its higher porosity compared to the bulk matrix which is related to the higher local water to cement ratio (Scrivener et al., 2004). The local increase in porosity leads to poorer bonds between the components, and thus to lower values of the mechanical fracture parameters of the ITZ, see e. g. Zacharda et al. (2018) or Vyhlídal and Klusák (2020). The properties of ITZ influence the overall fracture behavior of concrete. 2.2. Linear elastic fracture mechanics and its generalized form Fracture mechanics is a widely used tool for an assessment of crack behavior in materials. Linear elastic fracture mechanics supposes cracks in homogenous, isotropic, and linearly elastic material, see Anderson (2005). Williams (1957) presented a universal solution for the stress components around a crack tip in the form of infinite series. Near crack tip (𝑟𝑟 → 0), only the first (singular) term is used to describe the stress field, while the higher order terms of the infinite series can sometimes be neglected. Components of the stress tensor are given by the superposition of three basic failure modes defined by Irwin (1957), where the opening mode I is predominant failure mode. Generalized linear elastic fracture mechanics deals with an assessment of general singular stress concentrators, such as sharp notches, bi-material notches, bi-material inclusions, interface cracks etc. – see Klusák et al. (2013), De Corte et al. (2017), or Klusák et al. (2016). In most cases, the loading modes cannot be assigned to the series terms, so the terms are labeled by Arabic subscripts. The dependence of stress on polar coordinate r is expressed by a general stress singularity exponent p1: 𝜎𝜎 ij =𝐻𝐻1 √2𝜋𝜋 𝑟𝑟−𝑝𝑝1𝐹𝐹𝑖𝑖𝑖𝑖 1(  )    ሺͳሻ where σij are the stress tensor components for i, j = r,  . Generalized stress intensity factor H1 have the units [MPa𝑚𝑚𝑝𝑝1], and Fij are the shape functions. For cracks in homogeneous media, the stress intensity factors can be determined directly in the FEM software, e.g. Ansys Inc. software (2021), while for the general singular stress concentrators, various direct or integration methods are used, see Ping et al. (2008), Klusák et al. (2008) or Profant et al. (2008). The values H1 are ascertained from a numerical solution of the studied geometry, materials, and boundary conditions. 2.3. Criterion of stability based on average stress ahead of the crack tip In the paper, the crack propagation condition is determined from the generalized maximum tangential stress (GMTS) criterion. This stability condition is based on two well-known fracture mechanics stability conditions – (a) crack initiation occurs if the stress intensity factor KI reaches its critical value KIc (also known as fracture toughness), see e.g. Anderson (2005), and (b) crack will propagate in the direction where the tangential stress σθθ is maximal, see Erdogan and Sih (1963). The GMTS stability criterion employs the average stress σθθ(𝜃𝜃) calculated across a distance d ahead of the crack tip. The distance d is usually chosen in dependence on the mechanism of a rupture (dimension of a plastic zone of metals, dimension of process zone of quasi-brittle materials, material grain size, or finite supposed crack initiation/propagation increment). The average stress σθθ(𝜃𝜃) ahead of the crack tip is given by the following equation. 𝜎𝜎𝜃𝜃𝜃𝜃 =1 𝑑𝑑∫𝜎𝜎𝜃𝜃𝜃𝜃(𝑟𝑟,𝜃𝜃)𝑑𝑑𝑟𝑟 𝑑𝑑 0 (2) The crack propagation direction is supposed in the direction of maximum of the average tangential stress resulting from numerical analysis. The critical value of tangential stress σθθ(𝜃𝜃) corresponding to the crack initiation is determined from the knowledge of crack propagation conditions for a crack in homogeneous material under mode I, where fracture toughness KIc is employed: 𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 =2𝐾𝐾𝐼𝐼𝐼𝐼 √2𝜋𝜋𝑑𝑑 (3) Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 1003 Author name / Structural Integrity Procedia 00 (2019) 000–000 3 its higher porosity compared to the bulk matrix which is related to the higher local water to cement ratio (Scrivener et al., 2004). The local increase in porosity leads to poorer bonds between the components, and thus to lower values of the mechanical fracture parameters of the ITZ, see e. g. Zacharda et al. (2018) or Vyhlídal and Klusák (2020). The properties of ITZ influence the overall fracture behavior of concrete. 2.2. Linear elastic fracture mechanics and its generalized form Fracture mechanics is a widely used tool for an assessment of crack behavior in materials. Linear elastic fracture mechanics supposes cracks in homogenous, isotropic, and linearly elastic material, see Anderson (2005). Williams (1957) presented a universal solution for the stress components around a crack tip in the form of infinite series. Near crack tip (𝑟𝑟 → 0), only the first (singular) term is used to describe the stress field, while the higher order terms of the infinite series can sometimes be neglected. Components of the stress tensor are given by the superposition of three basic failure modes defined by Irwin (1957), where the opening mode I is predominant failure mode. Generalized linear elastic fracture mechanics deals with an assessment of general singular stress concentrators, such as sharp notches, bi-material notches, bi-material inclusions, interface cracks etc. – see Klusák et al. (2013), De Corte et al. (2017), or Klusák et al. (2016). In most cases, the loading modes cannot be assigned to the series terms, so the terms are labeled by Arabic subscripts. The dependence of stress on polar coordinate r is expressed by a general stress singularity exponent p1: 𝜎𝜎 ij =𝐻𝐻1 √2𝜋𝜋 𝑟𝑟−𝑝𝑝1𝐹𝐹𝑖𝑖𝑖𝑖 1(  )    ሺͳሻ where σij are the stress tensor components for i, j = r,  . Generalized stress intensity factor H1 have the units [MPa𝑚𝑚𝑝𝑝1], and Fij are the shape functions. For cracks in homogeneous media, the stress intensity factors can be determined directly in the FEM software, e.g. Ansys Inc. software (2021), while for the general singular stress concentrators, various direct or integration methods are used, see Ping et al. (2008), Klusák et al. (2008) or Profant et al. (2008). The values H1 are ascertained from a numerical solution of the studied geometry, materials, and boundary conditions. 2.3. Criterion of stability based on average stress ahead of the crack tip In the paper, the crack propagation condition is determined from the generalized maximum tangential stress (GMTS) criterion. This stability condition is based on two well-known fracture mechanics stability conditions – (a) crack initiation occurs if the stress intensity factor KI reaches its critical value KIc (also known as fracture toughness), see e.g. Anderson (2005), and (b) crack will propagate in the direction where the tangential stress σθθ is maximal, see Erdogan and Sih (1963). The GMTS stability criterion employs the average stress σθθ(𝜃𝜃) calculated across a distance d ahead of the crack tip. The distance d is usually chosen in dependence on the mechanism of a rupture (dimension of a plastic zone of metals, dimension of process zone of quasi-brittle materials, material grain size, or finite supposed crack initiation/propagation increment). The average stress σθθ(𝜃𝜃) ahead of the crack tip is given by the following equation. 𝜎𝜎𝜃𝜃𝜃𝜃 =1 𝑑𝑑∫𝜎𝜎𝜃𝜃𝜃𝜃(𝑟𝑟,𝜃𝜃)𝑑𝑑𝑟𝑟 𝑑𝑑 0 (2) The crack propagation direction is supposed in the direction of maximum of the average tangential stress resulting from numerical analysis. The critical value of tangential stress σθθ(𝜃𝜃) corresponding to the crack initiation is determined from the knowledge of crack propagation conditions for a crack in homogeneous material under mode I, where fracture toughness KIc is employed: 𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 =2𝐾𝐾𝐼𝐼𝐼𝐼 √2𝜋𝜋𝑑𝑑 (3) 4 Author name / Structural Integrity Procedia 00 (2019) 000–000 The crack propagates when the mean value of tangential stress σθθ(𝜃𝜃0) exceeds its critical value 𝜎𝜎𝜃𝜃𝜃𝜃,c . 𝜎𝜎𝜃𝜃𝜃𝜃(𝜃𝜃0)≥ 𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 (4) In the case of bi-material combinations, the crack propagates into direction θ0 and material m corresponding to the lowest value of critical applied force Fcrit,m. The critical applied force Fcrit,m depends on fracture toughness of particular material KIc,m and on the ratio of the critical value 𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 and the average tangential stress corresponding to the applied force Fappl. 𝐹𝐹crit,𝑚𝑚 = 𝐹𝐹appl 𝜎𝜎 𝜃𝜃𝜃𝜃,c(𝐾𝐾Ic,𝑚𝑚) 𝜎𝜎𝜃𝜃𝜃𝜃(𝜃𝜃0,𝑚𝑚) (5) 3. Numerical study 3.1. Numerical model A simplified 2D model of the specimen was created in Ansys Inc. software (2021). The dimensions of the specimen are in Fig. 1. The specimen contains a steel inclusion with dimensions 8×8×40 mm placed in the middle of the span. The specimens include an initial notch with a depth of 12 mm. Fig. 1. Designed specimen geometry (dimensions in mm). Materials were modelled as linear, elastic, and isotropic and they were represented by the elastic constants, i.e. Young’s modulus E, and Poisson’s ratio ν. Fracture properties were defined by the fracture toughness KIc (Table 1). Table 1. Overview of the material’s parameters used in the numerical model. Material E [GPa] ν [–] K Ic [MPa∙m1/2] Matrix (MTX) – fine-grained cement-based composite 39.3 0.20 0.58 ITZ 14.7 0.20 0.7K Ic, MTX Inclusion – steel S235 210 0.30 irrelevant Young’s modulus of elasticity of the ITZ EITZ was estimated according to approaches in Zacharda et al. (2018). Fracture toughness of the ITZ was chosen as 0.7KIc, MTX. The crack was modelled as ideally sharp. Because of high stress gradients surrounding crack tips and edges, highly refined meshes were required near the crack tip, bottom, left and upper corner of steel inclusion, see Fig. 2. In the following study of crack propagation, the resulting critical forces will be compared to a reference specimen. The reference specimen has the same geometry but is made of homogeneous material with the properties of matrix. In the graphs of results, the critical forces corresponding to the specimens with steel inclusion are labeled as STE, while Fcrit for the reference specimens are labeled as REF. 1004 Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 Author name / Structural Integrity Procedia 00 (2019) 000–000 5 Fig. 2. Radial mesh around – crack tip (left); left corner of steel inclusion (middle); top corner of steel inclusion (right). 3.2. Crack initiation from the notch Distance d was considered as the length between the crack tip and bottom corner of the inclusion, which are both potential stress concentrators. According to the theories of finite fracture mechanics, see Taylor et al. (2005), Cornetti et al. (2006), Taylor (2017), we suppose, that this distance d corresponds to current crack increment of final length. We suppose that the bridge between the initial crack (edge notch) and the inclusion breaks at once. The stress concentration ahead of the crack tip and the inclusion is shown in Fig. 3. Fig. 3. Isosurfaces of the first principal stress for the specimen with the steel inclusion (left) and the homogeneous reference specimen (right). Various notch depths would lead to different stress distributions as shown in Vyhlídal and Klusák (2019), where it was found that the diamond blade of the saw can damage the specimen more than expected. Here in the study of the notch with the depth 12 mm, the critical applied force was determined as crack propagation in the matrix: Fcrit (STE) = 1.36 kN. Similarly, the critical forces were calculated for the reference specimens: Fcrit (REF) = 1.14 kN. 3.3. Crack propagation from the bottom corner of the inclusion In this stage, we analyzed the crack propagation in the bottom corner of the inclusion. The distribution of the first principal stress can be seen in Fig. 4 (left), while a graph describing the development of the critical force Fcrit in dependence on the distance d is shown in Fig. 4 (right). It should be noted that the critical force Fcrit was determined for the crack propagation along the interface, whose fracture toughness was equal to 70 % of the fracture toughness of the matrix in the numerical model. The critical forces for reference specimen are also shown in Fig. 4 (right). Although the crack propagation along the interface is not natural, in contrast to the vertical crack propagation in the case of reference specimens, the Fcrit (STE) reaches lower values in comparison to the Fcrit (REF). It is caused by the lower fracture toughness of the interface KIc, ITZ compared to the KIc, MTX. Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 1005 Author name / Structural Integrity Procedia 00 (2019) 000–000 5 Fig. 2. Radial mesh around – crack tip (left); left corner of steel inclusion (middle); top corner of steel inclusion (right). 3.2. Crack initiation from the notch Distance d was considered as the length between the crack tip and bottom corner of the inclusion, which are both potential stress concentrators. According to the theories of finite fracture mechanics, see Taylor et al. (2005), Cornetti et al. (2006), Taylor (2017), we suppose, that this distance d corresponds to current crack increment of final length. We suppose that the bridge between the initial crack (edge notch) and the inclusion breaks at once. The stress concentration ahead of the crack tip and the inclusion is shown in Fig. 3. Fig. 3. Isosurfaces of the first principal stress for the specimen with the steel inclusion (left) and the homogeneous reference specimen (right). Various notch depths would lead to different stress distributions as shown in Vyhlídal and Klusák (2019), where it was found that the diamond blade of the saw can damage the specimen more than expected. Here in the study of the notch with the depth 12 mm, the critical applied force was determined as crack propagation in the matrix: Fcrit (STE) = 1.36 kN. Similarly, the critical forces were calculated for the reference specimens: Fcrit (REF) = 1.14 kN. 3.3. Crack propagation from the bottom corner of the inclusion In this stage, we analyzed the crack propagation in the bottom corner of the inclusion. The distribution of the first principal stress can be seen in Fig. 4 (left), while a graph describing the development of the critical force Fcrit in dependence on the distance d is shown in Fig. 4 (right). It should be noted that the critical force Fcrit was determined for the crack propagation along the interface, whose fracture toughness was equal to 70 % of the fracture toughness of the matrix in the numerical model. The critical forces for reference specimen are also shown in Fig. 4 (right). Although the crack propagation along the interface is not natural, in contrast to the vertical crack propagation in the case of reference specimens, the Fcrit (STE) reaches lower values in comparison to the Fcrit (REF). It is caused by the lower fracture toughness of the interface KIc, ITZ compared to the KIc, MTX. 6 Author name / Structural Integrity Procedia 00 (2019) 000–000 Fig. 4. Isosurfaces of the first principal stress (left), and a graph describing the dependence of the Fcrit on the value of d (right). 3.4. Crack propagation in the left corner of the inclusion The distribution of average tangential stress 𝜎𝜎𝜃𝜃𝜃𝜃 in dependence on the polar coordinate  and the distance d, for the crack with its tip at the left corner of steel inclusion can be seen in Fig. 5 (right). The polar coordinate  between 0 ° and 90 ° represents the area of the steel inclusion which means, that the direction  = 0° is oriented in the edge of the inclusion (crack face) and positive values are counted counterclockwise. The graph clearly describes the maximum 𝜎𝜎𝜃𝜃𝜃𝜃 in the direction about 135°, which goes to the matrix (almost vertically). This is also in the agreement with the distribution of the first principal stress (Fig. 5, left). Fig. 5. Isosurfaces of the first principal stress (left), Average opening stress 𝜎𝜎𝜃𝜃𝜃𝜃 (right). Nevertheless, to determine the crack propagation direction, critical applied forces must be determined separately for each material region. Of course, propagation to the steel inclusion is not possible, and only the matrix or the ITZ can be considered as potential crack propagation regions. These two regions are characterized by the values of fracture toughness KIc,MTX, and KIc, ITZ. The graph of Fcrit is shown in Fig. 6 in dependence on the material regions and on the distance d. We can see that the minimum value of critical force is in the ITZ, because we have chosen the fracture toughness of ITZ as 0.7 × KIc,MTX. If fracture toughness of ITZ is greater, crack propagates vertically to the matrix, if it is less than or equal to 0.7  KIc,MTX, crack propagates along the interface between steel inclusion and the matrix. Note that also the distance d plays a role in the competition between the crack propagation direction into MTX or ITZ. The comparison of critical forces of specimen with the inclusion and the reference specimen is not easy here, because the crack propagates in a straight line in homogeneous material. For this reason, the values of Fcrit (REF) are calculated for the crack tip in the center of the specimen. Due to the lower values of the fracture toughness of the ITZ KIc, ITZ, the Fcrit (STE) also reaches lower values in comparison to the Fcrit (REF). 1006 Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 Author name / Structural Integrity Procedia 00 (2019) 000–000 7 Fig. 6. Distribution of critical force Fcrit in dependence on the polar coordinate  , and in dependence on the distance d (on the left); minimum value of critical force Fcrit in dependence on the distance d (on the right). 3.5. Crack propagation from the top corner of the inclusion The graph in Fig. 7 shows the distribution of average tangential stress for the crack with its tip at the top corner of steel inclusion in dependence on the polar coordinate and distance d. Crack propagation in the almost vertical direction (about 215°) is expected. Further, the crack will turn in vertical direction. The dependences of critical forces Fcrit on the distance d are shown in Fig. 7 (right) for the specimens both with and without the inclusion. Similarly to the section 3.4, the values of Fcrit (REF) are calculated for the crack tip in the position which corresponds to the top corner of the inclusion. The differences between Fcrit (REF) and Fcrit (STE) are due to the different directions of crack propagation for both specimens (STE and REF) but are negligible in this case. Fig. 7. Distribution of average values of tangential stress 𝜎𝜎𝜃𝜃𝜃𝜃 in dependence on polar angle (potential crack propagation direction), critical force Fcrit in dependence on the distance d, respectively. 4. Conclusions The numerical study in the previous paragraphs estimated potential crack propagation paths and the critical applied forces during crack propagation stages: crack initiation in the central edge notch, and propagation in the bottom, left, and upper corners of the inclusion. The Fig. 8 shows the evolution of the critical force Fcrit in the crack stages. The results of critical forces for crack propagation around the inclusion are compared to the crack propagation in homogeneous material (the reference specimen). Although the dependence on the distance d is often discussed, it is apparent, that it is very weak. Crack propagation path depends mainly on the fracture properties of ITZ, see the section 3.4. The comparison of the specimens with and without the inclusion shows that the central inclusion works as an obstacle to crack propagation only in the beginning stage – the crack initiation in the notch tip, where Fcrit (STE) is higher than Fcrit (REF). The mechanical fracture parameters of the ITZ strongly influence the overall fracture behavior of test specimens, as we reported in previous papers, see e.g. Vyhlídal et al. (2022), Vyhlídal and Klusák (2020) or Zacharda et al. (2018). 8 Author name / Structural Integrity Procedia 00 (2019) 000–000 Fig. 8. Evolution of the critical applied force Fcrit for the crack propagation stages in dependence on the distance d. Acknowledgements This outcome has been achieved with the financial support of the Brno University of Technology under project No. FAST-J-22-8038. Authors are grateful for support from the Czech Science Foundations, project No. 21-08772S. References Anderson, T. L., 2005. Fracture mechanics: fundamentals and applications (3rd ed). CRC Press, Boca Raton, FL, pp. 630. ANSYS, Inc. ANSYS 2021 R2 Users Guide. Swanson Analysis System, Houston, 2021. Cornetti P., Pugno N., Carpinteri A., Taylor D., 2006. Finite fracture mechanics: a coupled stress and energy failure criterion. Eng. Fract. Mech. 73, 2021–33. De Corte, W., Helincks, P., Boel, V., Klusák, J., Seitl, S., De Schutter, G. 2017. Generalised fracture mechanics approach to the interfacial failure analysis of a bonded steel-concrete joint, Frattura ed Integrita Strutturale 42, 147-160. Diamond, S., Huang, J., 1998. Interfacial Transition Zone: Reality or Myth? In: The Interfacial Transition Zone in Cementitious Composites. E&FN Spon, London, pp. 3-39. EN 1992-1-1. Eurocode 2: Design of concrete structures: Part 1-1: General rules and rules for buildings. 2. Brussel: European Committee for Standardization, 2011. Erdogan, F., Sih, G. C., 1963. On the Crack Extension in Plates Under Plane Loading and Transverse Shear. Journal of Basic Eng. 85, 519–525. Farran, J., 1956. Contribution mineralogique a l’etude de l’adherence entre les constituants hydrates des ciments et les materiaux enrobes. Revue des Matiriaux de Construction 491, 155–157. Irwin, G., 1957. Analysis of Stresses and Strains near the End of a Crack Traversing a Plate. Journal of Applied Mechanics 24, 361–364. Klusák, J., Profant, T., Kotoul, M., 2008. A comparison of two direct methods of generalized stress intensity factor calculations of bi-material notches, Key Eng. Mater. 385–387, 409–412. Klusák, J., Profant, T., Knésl, Z., Kotoul, M. 2013. The influence of discontinuity and orthotropy of fracture toughness on conditions of fracture initiation in singular stress concentrators, Engineering Fracture Mechanics, 110, 438-447. Klusák, J., Krepl, O., Profant, T., 2016. Behaviour of a crack in a corner or at a tip of a polygon-like particle, Proc. Struct. Integ., 1912-1919. Ping, X.C., Chen, M.C., Xie, J.L., 2008. Singular stress analyses of V-notched anisotropic plates based on a novel finite element method. Eng. Fract. Mech. 75, 3819–3838 Profant, T., Ševeček, O., Kotoul, M., 2008. Calculation of K-factor and T-stress for cracks in anisotropic bimaterials, Eng. Fract. Mech. 75, 3707– 26. Scrivener, K. L., Crumbie, A. K., Laugesen, P., 2004. The interfacial transition zone (ITZ) between cement paste and aggregate in concrete. Interface Science 12(4), 411–421. Taylor, D., Cornetti, P., Pugno, N., 2005: The fracture mechanics of finite crack extension Engineering Fracture Mechanics, 72, 1021-1038. Taylor, D., 2017: The Theory of Critical Distances: A link to micromechanisms, Theoretical and Applied Fracture Mechanics, 90, 228-233. Vyhlídal, M., Klusák, J., 2019 The influence of polygonal cavity on fracture behaviour of concrete. Procedia Structural Integrity. 690–697. Vyhlídal, M., Rozsypalová, I., Majda, T., Daněk, P., Šimonová, H., Kucharczyková, B., Keršner, Z., 2019. Fracture response of fine-grained cement-based composite specimens with special inclusions. Solid State Phenomena 292, pp. 63–68. Vyhlídal, M., Klusák, J., 2020. A Crack Approaching the Edge of the Aggregate. Transactions of the VŠB – Technical University of Ostrava, Civil Engineering Series 20 (2), 47−52. Vyhlídal, M., Rozsypalová, I., Šimonová, H., Kucharczyková, B., Vavro, L., Vavro, M., Němeček, J., Rovnaníková, P., Keršner, Z., 2022. Effect of petrographic composition and chemistry of aggregate on the local and general fracture response of cementitious composites. Frattura ed Integrità Strutturale 16 (60), 13–29. Williams, M. L., 1956. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24, 109–114. Zacharda, V., Němeček, J., Šimonová, H., Kucharczyková, B., Vyhlídal, M., Keršner, Z., 2018. Influence of interfacial transition zone on local and overall fracture response of cementitious composites. Key Engineering Materials 784, 97−102. Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 1007 Author name / Structural Integrity Procedia 00 (2019) 000–000 7 Fig. 6. Distribution of critical force Fcrit in dependence on the polar coordinate  , and in dependence on the distance d (on the left); minimum value of critical force Fcrit in dependence on the distance d (on the right). 3.5. Crack propagation from the top corner of the inclusion The graph in Fig. 7 shows the distribution of average tangential stress for the crack with its tip at the top corner of steel inclusion in dependence on the polar coordinate and distance d. Crack propagation in the almost vertical direction (about 215°) is expected. Further, the crack will turn in vertical direction. The dependences of critical forces Fcrit on the distance d are shown in Fig. 7 (right) for the specimens both with and without the inclusion. Similarly to the section 3.4, the values of Fcrit (REF) are calculated for the crack tip in the position which corresponds to the top corner of the inclusion. The differences between Fcrit (REF) and Fcrit (STE) are due to the different directions of crack propagation for both specimens (STE and REF) but are negligible in this case. Fig. 7. Distribution of average values of tangential stress 𝜎𝜎𝜃𝜃𝜃𝜃 in dependence on polar angle (potential crack propagation direction), critical force Fcrit in dependence on the distance d, respectively. 4. Conclusions The numerical study in the previous paragraphs estimated potential crack propagation paths and the critical applied forces during crack propagation stages: crack initiation in the central edge notch, and propagation in the bottom, left, and upper corners of the inclusion. The Fig. 8 shows the evolution of the critical force Fcrit in the crack stages. The results of critical forces for crack propagation around the inclusion are compared to the crack propagation in homogeneous material (the reference specimen). Although the dependence on the distance d is often discussed, it is apparent, that it is very weak. Crack propagation path depends mainly on the fracture properties of ITZ, see the section 3.4. The comparison of the specimens with and without the inclusion shows that the central inclusion works as an obstacle to crack propagation only in the beginning stage – the crack initiation in the notch tip, where Fcrit (STE) is higher than Fcrit (REF). The mechanical fracture parameters of the ITZ strongly influence the overall fracture behavior of test specimens, as we reported in previous papers, see e.g. Vyhlídal et al. (2022), Vyhlídal and Klusák (2020) or Zacharda et al. (2018). 8 Author name / Structural Integrity Procedia 00 (2019) 000–000 Fig. 8. Evolution of the critical applied force Fcrit for the crack propagation stages in dependence on the distance d. Acknowledgements This outcome has been achieved with the financial support of the Brno University of Technology under project No. FAST-J-22-8038. Authors are grateful for support from the Czech Science Foundations, project No. 21-08772S. References Anderson, T. L., 2005. Fracture mechanics: fundamentals and applications (3rd ed). CRC Press, Boca Raton, FL, pp. 630. ANSYS, Inc. ANSYS 2021 R2 Users Guide. Swanson Analysis System, Houston, 2021. Cornetti P., Pugno N., Carpinteri A., Taylor D., 2006. Finite fracture mechanics: a coupled stress and energy failure criterion. Eng. Fract. Mech. 73, 2021–33. De Corte, W., Helincks, P., Boel, V., Klusák, J., Seitl, S., De Schutter, G. 2017. Generalised fracture mechanics approach to the interfacial failure analysis of a bonded steel-concrete joint, Frattura ed Integrita Strutturale 42, 147-160. Diamond, S., Huang, J., 1998. Interfacial Transition Zone: Reality or Myth? In: The Interfacial Transition Zone in Cementitious Composites. E&FN Spon, London, pp. 3-39. EN 1992-1-1. Eurocode 2: Design of concrete structures: Part 1-1: General rules and rules for buildings. 2. Brussel: European Committee for Standardization, 2011. Erdogan, F., Sih, G. C., 1963. On the Crack Extension in Plates Under Plane Loading and Transverse Shear. Journal of Basic Eng. 85, 519–525. Farran, J., 1956. Contribution mineralogique a l’etude de l’adherence entre les constituants hydrates des ciments et les materiaux enrobes. Revue des Matiriaux de Construction 491, 155–157. Irwin, G., 1957. Analysis of Stresses and Strains near the End of a Crack Traversing a Plate. Journal of Applied Mechanics 24, 361–364. Klusák, J., Profant, T., Kotoul, M., 2008. A comparison of two direct methods of generalized stress intensity factor calculations of bi-material notches, Key Eng. Mater. 385–387, 409–412. Klusák, J., Profant, T., Knésl, Z., Kotoul, M. 2013. The influence of discontinuity and orthotropy of fracture toughness on conditions of fracture initiation in singular stress concentrators, Engineering Fracture Mechanics, 110, 438-447. Klusák, J., Krepl, O., Profant, T., 2016. Behaviour of a crack in a corner or at a tip of a polygon-like particle, Proc. Struct. Integ., 1912-1919. Ping, X.C., Chen, M.C., Xie, J.L., 2008. Singular stress analyses of V-notched anisotropic plates based on a novel finite element method. Eng. Fract. Mech. 75, 3819–3838 Profant, T., Ševeček, O., Kotoul, M., 2008. Calculation of K-factor and T-stress for cracks in anisotropic bimaterials, Eng. Fract. Mech. 75, 3707– 26. Scrivener, K. L., Crumbie, A. K., Laugesen, P., 2004. The interfacial transition zone (ITZ) between cement paste and aggregate in concrete. Interface Science 12(4), 411–421. Taylor, D., Cornetti, P., Pugno, N., 2005: The fracture mechanics of finite crack extension Engineering Fracture Mechanics, 72, 1021-1038. Taylor, D., 2017: The Theory of Critical Distances: A link to micromechanisms, Theoretical and Applied Fracture Mechanics, 90, 228-233. Vyhlídal, M., Klusák, J., 2019 The influence of polygonal cavity on fracture behaviour of concrete. Procedia Structural Integrity. 690–697. Vyhlídal, M., Rozsypalová, I., Majda, T., Daněk, P., Šimonová, H., Kucharczyková, B., Keršner, Z., 2019. Fracture response of fine-grained cement-based composite specimens with special inclusions. Solid State Phenomena 292, pp. 63–68. Vyhlídal, M., Klusák, J., 2020. A Crack Approaching the Edge of the Aggregate. Transactions of the VŠB – Technical University of Ostrava, Civil Engineering Series 20 (2), 47−52. Vyhlídal, M., Rozsypalová, I., Šimonová, H., Kucharczyková, B., Vavro, L., Vavro, M., Němeček, J., Rovnaníková, P., Keršner, Z., 2022. Effect of petrographic composition and chemistry of aggregate on the local and general fracture response of cementitious composites. Frattura ed Integrità Strutturale 16 (60), 13–29. Williams, M. L., 1956. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24, 109–114. Zacharda, V., Němeček, J., Šimonová, H., Kucharczyková, B., Vyhlídal, M., Keršner, Z., 2018. Influence of interfacial transition zone on local and overall fracture response of cementitious composites. Key Engineering Materials 784, 97−102.